Exercise 3.3

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MA111: Prepared by Dr.Archara Pacheenburawana 26
Exercise 3.3
1. For each of the numbers a, b, c, d, e, r, s, and t, state whether the function whose
graphisshown hasanabsolutemaximum orminimum, a localmaximum orminimum,
or neither a maximum nor a minimum.
(a)
y
a b c d e r s t
x
(b)
y
a b c d e r s
t
x
2. Use the graph to state the absolute and local maximum and minimum values of the
function.
(a)
y
y = f(x)
1
0
1
x
(b)
y
y = f(x)
1
0
1
x

MA111: Prepared by Dr.Archara Pacheenburawana 27
3. Find all critical numbers and the local extreme values of the following functions.
(a) f(x) = x 2 +5x−1
(b) f(x) = x 3 −3x+1
(c) f(x) = x 3 −3x 2 +3x (d) f(x) = x 4 −3x 3 +2
(e) f(x) = x 3/4 −4x 1/4 (f) f(x) = x 3 −2x 2 −4x
(g) f(x) = sinxcosx, [0,2π] (h) f(x) = x+1
x−1
(i) f(x) = x (j) f(x) = 1
x 2 2
+1
(ex +e −x )
(k) f(x) = x 4/3 +4x 1/3 +3x −2/3 (l) f(x) = 2x √ x+1
(m) f(x) = e −x2
(n) f(x) = sinx 2 , [0,π]
4. Find the absolute maximum and absolute minimum values of f on the given interval.
(a) f(x) = 3x 2 −12x+5, [0,3]
(b) f(x) = 2x 3 +3x 2 +4, [−2,1]
(c) f(x) = x 4 −4x 2 +2, [−3,2]
(d) ) f(x) = x 2 − 2 x , [1 2 ,2]
(e) f(x) = x
x 2 +1 , [0,2]
(f) f(x) = sinx+cosx, [0,π/3]
(g) f(x) = xe −x , [0,2]
(h) f(x) = x−3lnx, [1,4]
(i) f(x) = |x−1|, [0,3]
(j) f(x) = 3√ x, [−1,27]
Answer to Exercise 3.3
1. (a) Absolute maximum at b; local maxima at b, e, and r;
absolute minimum at d; local minima at d and s
(b) Absolute maximum at e; local maxima at e and s;
absolute minimum at t; local minima at b, c, d, r, and t
2. (a) Absolute maximum f(4) = 4; absolute minimum f(7) = 0
local maximum f(4) = 4, f(6) = 3; local minimum f(2) = 1, f(5) = 2
(b) Absolute maximum f(7) = 5; absolute minimum f(1) = 0
local maximum f(0) = 2, f(3) = 4, f(5) = 3
local minimum f(1) = 0, f(4) = 2, f(6) = 1

MA111: Prepared by Dr.Archara Pacheenburawana 28
3. (a) − 5 , absolute minimum (b) −1, local maximum; 1, local minimum
2
(c) 1, no local extreme values
9
(d) 0, no local extreme values; , local minimum
4
16
(e) 0, no local extreme values; , local minimum
9
(f) − 2 , local minimum; 2, local maximum
3
(g) π, 5π, local maximum; 3π
, 7π , local minimum
4 4 4 4
(h) 1, no local extreme values
(i) −1, local minimum; 1, local maximum
(j) 0, local minimum (k) −2,1, local minimum
(l) − 2 , local minimum (m) 0, local maximum
3
√
3π
(n) 0, , local minimum; √ √
π
, 5π, local maximum
2 2 2
4. (a) f(0) = 5,f(2) = −7 (b) f(1) = 9,f(−2) = 0 (c) f(−3) = 47,f(± √ 2) = −2
(d) f(2) = 3,f(1) = −1 (e) f(1) = 1 2 ,f(0) = 0 (f) f(π/4) = √ 2,f(0) = 1
(g) f(1) = 1/e,f(0) = 0 (h) f(1) = 1,f(3) = 3−3ln3 (i) f(3) = 2,f(1) = 0
(j) f(27) = 3,f(−1) = −1
Exercise 3.4
1. Find the intervals on which f is increasing or decreasing.
(a) f(x) = x 3 −3x+2 (b) f(x) = x 4 −8x 2 +1
(c) f(x) = x 6 +192x+17 (d) f(x) = (x+1) 2/3
(e) f(x) = x−2sinx, [0,2π]
(g) f(x) = e x2 −1
(f) f(x) = sin3x, [0,π]
(h) f(x) = (lnx)/ √ x
2. At what values of x does f have a local maximum or minimum Sketch the graph.
(a) f(x) = x 3 +2x 2 −x−1 (b) f(x) = x √ x 2 +1
(c) f(x) = xe −2x (d) f(x) = lnx 2
(e) f(x) = x (f) f(x) = x3
x 2 −1
x 2 −1
(g) f(x) = sinx+cosx (h) f(x) = √ x 3 +3x 2
(i) f(x) = x 2/3 −2x −1/3
3. Show that 5 is a critical number of g(x) = 2 + (x − 5) 3 but g has no local extreme
values at 5.
4. Find a cubic function f(x) = ax 3 +bx 2 +cx+d that has a local maximum value of 3
at −2 and a local minimum value of 0 at 1.

MA111: Prepared by Dr.Archara Pacheenburawana 29
Answer to Exercise 3.4
1. (a) Inc. on (−∞,−1)∪(1,∞); dec. on (−1,1)
(b) Inc. on (−2,0)∪(2,∞); dec. on (−∞,−2)∪(0,2)
(c) Inc. on (−2,∞); dec. on (−∞,−2) (d) Inc. on (−1,∞); dec. on (−∞,−1)
(e) Inc. on ( π , 5π 3 3 )∪(7π,3π); dec. on (0, π 3 3 )∪(5π, 7π ) 3 3
(f) Inc. on (0, π 6 )∪(3π, 5π); dec. on 6 6 (π, 3π 6 6 )∪(5π,π)
6
(g) Inc. on (0,∞); dec. on (−∞,0) (h) Inc. on (0,e 2 ); dec. on (e 2 ,∞)
2. (a) Loc. max. at x = − 2 − √ 7
; loc. min. at x = 3 3 −2 + √ 7
3 3
y
2
2
x
(b) None
10
y
2
x
(c) Loc. max. at x = 1 2
y
0.2
1
x
(d) None
y
1
1
x

MA111: Prepared by Dr.Archara Pacheenburawana 30
(e) None
y
2
1
x
(f) Loc. max. at x = − √ 3; Loc. min. at x = √ 3
y
2
4
x
(g) Loc. max. at x = π +2nπ; Loc. min. at x = 5π +2nπ
4 4
y
2
π
x
(h) Loc. max. at x = −2; Loc. min. at x = 0
y
(i) Loc. min. at x = −1
2
y
1
x
2
2
x
4. f(x) = 1 9 (2x3 +3x 2 −12x+7)

MA111: Prepared by Dr.Archara Pacheenburawana 31
Exercise 3.5
1. Use the given graph of f to find the intervals of concavity.
(a)
y
20
10
−3
−1
1 3
x
(b)
y
1
−4
−2
2 4
x
(c)
y
10
5
−2
2 4
x
−5
−10
(d)
y
10
5
−3
−1
1 2 3
x
−5
2. Given the following functions.
• Find the interval of increase and decrease.
• Find the local maximum and minimum values.
• Find the intervals of concavity and the inflection points.

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MA111: Prepared by Dr.Archara Pacheenburawana 32
• Use the above information to sketch the graph.
(a) f(x) = 2x 3 −3x 2 −12x (b) f(x) = x 4 −6x 2
(c) f(x) = 3x 5 −5x 3 +3 (d) f(x) = x √ x 2 +1
(e) f(x) = x 1/3 (x+3) 2/3 (f) f(x) = sin 2 x
Answer to Exercise 3.5
1. (a) CU on (−∞,−1)∪(1,∞); CD on (−1,1)
(b) CU on (−∞,0); CD on (0,∞)
(c) CU on (1,∞); CD on (−∞,1)
(d) CU on (−1,0)∪(1,∞); CD on (−∞,−1)∪(0,1)
2. (a) Inc. on (−∞,−1)∪(2,∞); dec. on (−1,2); loc. max. f(−1) = 7;
loc. min. f(2) = −20; CU on ( 1,∞); CD on (−∞, 1); IP 2 2 (1 2 ,−13)
2
y
−10
5
1
x
−20
(b) Inc. on (− √ 3,0)∪( √ 3,∞); dec. on (−∞,− √ 3)∪(0, √ 3);
loc. min. f(± √ 3) = −9; loc. max. f(0) = 0; CU on (−∞,−1)∪(1,∞);
CD on (−1,1); IP (±1,−5)
y
−1
1
• •
x
• •
−10
(c) Inc. on (−∞,−1)∪(1,∞); dec. on (−1,−1); loc. max. f(−1) = 5;
loc. min. f(1) = 1; CU on (−1/ √ 2,0)∪(1/ √ 2,∞);
CD on (−∞,−1/ √ 2)∪(1/ √ 2,∞); IP (0,3), (±1/ √ 2,3∓ 8√ 7 2)

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MA111: Prepared by Dr.Archara Pacheenburawana 33
y
5
−1
1
x
(d) Inc. on (−∞,∞); No loc. max and loc. min; CU on (0,∞);
CD on (−∞,0); IP (0,0)
y
x
e) Inc. on (−∞,−3)∪(−1,∞); dec. on (−3,−1); loc. max. f(−3) = 0;
loc. min. f(−1) = − 3√ 4; CU on (−∞,−3)∪(−3,0); CD on (0,∞); IP (0,0)
y
2
1
x
(f) Inc. on (0,π/2)∪(π,3π/2); dec. on (π/2,π)∪(3π/2,2π)
loc. max. f(π/1) = f(3π/2) = 1; loc. min. f(π) = 0
CU on (0,π/4)∪(3π/4,5π/4)∪(7π/4,2π); CD on (π/4,3π/4)∪(5π/4,7π/4)
IP (π/4, 1),(3π/4, 1),(5π/4, 1),(7π/4, 1)
2 2 2 2
y
1
0
π
2
π
3π
2
2π
x

MA111: Prepared by Dr.Archara Pacheenburawana 34
Exercise 3.6
1. Find two positive numbers whose product is 100 and whose sum is minimum.
2. Find two numbers whose product is −16 and the sum of whose squares is minimum.
3. Forwhatnumber doestheprincipal fourthrootexceed twicethenumberbythelargest
amount.
4. Find the dimensions of a rectangle with area 1000 m 2 whose perimeter is as small as
possible.
5. A box with a square base and open top must have the volume of 32,000 cm 3 . Find
the dimensions of the box that minimize the amount of material used.
6. If 1200 cm 2 of material is available to make a box with a square base and an open
top, find the largest possible volume of the box.
7. A farmer wishes to fence off two identical adjoining rectangular pens, each with 900
square feet of area, as shown in the following Figure.
y
x
What are x and y so that the least amount of fence is required
8. Find the point on the line y = 4x+7 that is closest to the origin.
9. Find the point on the line 6x+y = 9 that is closest to the point (−3,1).
10. Find the point on the parabola y = x 2 that is closest to the point (0,5).
11. Find the point on the parabola x+y 2 = 0 that is closest to the point (0,−3).
12. Find the dimensions of the rectangle of largest area that can be inscribed in a circle
of radius r.
13. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a
circle of radius r.
14. A right circular cylinder is inscribed in a cone with height h and base radius r. Find
the largest possible volume of such a cylinder.
15. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible
surface area of such a cylinder.
16. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible
volume of such a cylinder.
17. Show that the rectangle with maximum perimeter that can be inscribed in a circle is
a square.

MA111: Prepared by Dr.Archara Pacheenburawana 35
18. (a) Show that of all the rectangles with a given area, the one with smallest perimeter
is a square.
(b) Show that of all the rectangles with a given perimeter, the one with largest area
is a square.
19. At 7 : 00am. one ship was 60 miles due east from a second ship. If the first ship
sailed west at 20 miles per hour and the second ship sailed southeast at 30 miles per
hour, when were they closest together
20. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and
the other is bent into an equilateral triangle. How should the wire be cut so that the
total area enclosed is (a) a maximum (b) A minimum
Answer to Exercise 3.6
1. 10,10 2. −4,4 3.
1
16
4. 10 √ 10×10 √ 10 5. 40×40×20 6. 4,000 cm 3
7. 15 √ 3×20 √ 3 8. ( − 28,
)
7
17 17
9. ( − 45,
)
63
37 37
10.
( ) (
−√ 3
2
, 9 2
,
3√
2
, 9 2
11. (−1,−1) 12. √ 2r × √ 2r 13. Base √ 3r, height 3r/2 14.
4
27 πr2 h
15. πr 2( 1+ √ 5 ) 16. 4πr 3 /3 √ 3 19. 8 : 09am.
20. (a) Use all of the wire for the square (b) 40 √ 3/ ( 9+4 √ 3 ) m for the square
Exercise 3.7
1. Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given
interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.
(a) f(x) = x 2 −4x+1, [0,4] (b) f(x) = x 3 −3x 2 +2x+5, [0,2]
(c) f(x) = sin2πx, [−1,1]
(d) f(x) = x √ x+6, [−6,0]
2. Use the graph of f to estimate the values of c that satisfy the conclusion of the Mean
Value Theorem for the interval [0,8].
y
)
y = f(x)
1
1
x

MA111: Prepared by Dr.Archara Pacheenburawana 36
3. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the
given interval. Then find all numbers c that satisfy the conclusion of the Mean Value
Theorem.
(a) f(x) = 3x 2 +2x+5, [−1,1] (b) f(x) = x 3 −x, [0,2]
(c) f(x) = e −2x , [0,3] (d) f(x) = x
x+2 , [1,4]
Answer to Exercise 3.7
1. (a) 2 (b) 3±√ 3
3
(c) ± 1 4 ,±3 4
(d) −4 2. 0.8, 3.2, 4.4, 6.1
3. (a) 0 (b) 2 √
3
(c) − 1 2 ln[(1−e−6 )/6] (d) 3 √ 2−2
Exercise 4.1
1. Find the most general antiderivative of the function.
(a) f(x) = 5 (b) f(x) = x 2 +π
(c) f(x) = x 5/4 (d) f(x) = 1/ 3√ x 2
(e) f(x) = x 2 −x (f) f(x) = 4x 5 −x 3
(g) f(x) = 27x 7 +3x 5 −45x 3 + √ 2x
(h) f(x) = 3 x 2 − 2 x 3 (i) f(x) = 4x6 +3x 4
2. Evaluate the integral and check your answer by differentiating.
∫
∫
(a) 3x 4 dx
(b) (x 2 +x)dx
∫
∫
(c) (3x 4 −3x)dx (d) (x+1) 2 dx
∫
(e) 3 √ ∫
xdx (f)
(3− 1 )
dx
x
∫ 4 x 1/3 ∫
−3
(x 2 +1) 2
(g) dx (h) √ dx
x 2/3 x
∫
∫
(i) (sinx−cosx)dx (j) 2secxtanxdx
∫
∫
(k) 5sec 2 xdx (l) (3e x −2)dx
∫
∫
(m) (3cosx−1/x)dx (n)
(5x− 3 )
dx
e
∫ x e x ∫
+3
(o) dx (p) x 1/4 (x 5/4 −4)dx
e x
x 3

MA111: Prepared by Dr.Archara Pacheenburawana 37
Answer to Exercise 4.1
1. (a) 5x+C (b) 1 3 x3 +πx+C (c) 4 9 x9/4 +C (d) 3 3√ x+C (e) 1 3 x3 − 1 2 x2 +C
(f) 2 3 x6 − 1 4 x4 +C (g) 27
8 x8 + 1 2 x6 − 45 4 x4 + √ 2
2 x2 +C (h) − 3 x + 1 x 2 +C
(i) x 4 + 3 2 x2 +C
2. (a) 3 5 x5 +C (b) 1 3 x3 + x2
2 +C (c) 3 5 x5 − 3 2 x2 +C (d) 1 3 (x+1)3 +C
(e) 2x 3/2 +C (f) 3x+ 1 3 x−3 +C (g) 3 2 x2/3 −9x 1/3 +C (h) 2 9 x9/2 + 4 5 x5/2 +2x 1/2 +C
(i) −cosx−sinx+C (j) 2secx+C (k) 5tanx+C (l) 3e x −2x+C
(m) 3sinx−ln|x|+C (n) 5 2 x2 +3e −x +C (o) x−3e −x +C (p) 2 5 x5/2 − 16 5 x5/4
Exercise 4.2
1. Evaluate the integral by making the given substitution.
∫
(a) cos3xdx, u = 3x
∫
(b) x 2 (x 3 +2)dx, u = x 3 +2
∫
4
(c) dx, u = 1+2x
(1+2x)
3
∫ √ ( x+2)
3
(d) √ dx, u = √ x+2 x
2. Evaluate the indefinite integral
∫
(a) 2x(x 2 +3) 4 dx
∫ √x−1dx
(c)
∫
(e)
∫
(g)
dx
5−3x
1+4x
√ dx
1+x+2x
2
∫
(i) cos2xdx
∫
(k) xsin(x 2 )dx
∫
(m) cos 4 xsinxdx
∫
(o) secxtanx √ 1+secxdx
∫
(b) (2x+1)(x 2 +x) 3 dx
∫
x 2
(d) √
x3 −2 dx
∫
(f)
∫
(h)
2x+1
x 2 +x−1 dx
1
√ √ dx x( x+1)
2
∫
(j) cosx √ sinx+1dx
∫
sinx
(l) √ dx cosx
∫
(n) sinx(cosx+3) 3/4 dx
∫
(p) cosxe sinx dx

MA111: Prepared by Dr.Archara Pacheenburawana 38
∫
(q) e x√ ∫
1+e x dx (r) xe x2 +1 dx
∫
∫
dx
4
(s)
(t)
xlnx
x(lnx+1) dx 2
∫ √cotx
∫
(u) csc 2 xdx (v) sinx(cosx−1) 3 dx
∫
∫ e
(w) sec 3 x −e −x
xtanxdx (x) dx
e x +e−x ∫ ∫ 1+x
2x+3
(y)
1+x dx (z) 2 x+7 dx
Answer to Exercise 4.2
1. (a) 1 3 sin3x+C (b) 2 3 (x3 +2) 3/2 +C (c) −1/(1+2x) 2 +C (d) 1 2 (√ x+2) 4 +C
2. (a) 1 5 (x2 +3) 5 +C (b) 1 4 (x2 +x) 4 +C (c) 2 3 (x−1)3/2 +C (d) 2 3√
x3 −2+C
(e) − 1 3 ln|5−3x|+C (f) ln|x2 +x−1|+C (g) 2 √ 1+x+2x 2 +C
(h) −2( √ x+1) −1 +C (i) 1 2 sin2x+C (j) 2 3 (sinx+1)3/2 +C (k) − 1 2 cos(x2 )+C
(l) −2 √ cosx+C (m) − 1 5 cos5 x+C (n) − 4 7 (cosx+3)7/4 +C
(o) 2 3 (1+secx)3/2 +C (p) e sinx +C (q) 2 3 (1+ex ) 3/2 +C (r) 1 2 ex2 +1 +C
(s) ln|lnx|+C (t) −4(lnx+1) −1 +C (u) − 2 3 (cotx)3/2 +C (v) − 1 4 (cosx−1)4 +C
(w) 1 3 sec3 x+C (x) ln(e x +e −x )+C (y) tan −1 x+ 1 2 ln(1+x2 )+C
(z) 2(x+7)−11ln|x+7|+C
Exercise 4.3
1. Express the limit as a definite integral on the given interval.
(a) lim
n→∞
n∑
i=1
x i sinx i △x, [0,π]
n∑ e x i
(b) lim △x, [1,5]
n→∞ 1+x
i=1 i
(c) lim
n∑
n→∞
i=1
n∑
n→∞
(d) lim
i=1
[2(x ∗ i) 2 −5x ∗ i]△x, [0,1]
√
x
∗
i △x, [1,4]

MA111: Prepared by Dr.Archara Pacheenburawana 39
2. Use the form of the definition of the integral given in (4.7) to evaluate the integral.
(a)
(c)
(e)
∫ 1
0
∫ 5
1
∫ 5
0
3. Prove that
4. Prove that
2xdx
(2+3x−x 2 )dx
(1+2x 3 )dx
∫ b
a
∫ b
a
xdx = b2 −a 2
.
2
x 2 dx = b3 −a 3
.
3
(b)
(d)
(f)
∫ 5
−1
∫ 2
0
∫ 2
1
(1+3x)dx
(2−x 2 )dx
x 3 dx
5. Write the given sum or difference as a single integral in the form ∫ b
f(x)dx. a
6. If
7. If
(a)
(c)
(e)
1. (a)
∫ 8
2
∫ 1
∫ 2
0
∫ 2
0
∫ 3
f(x)dx+
f(x)dx+
f(x)dx+
∫ 3
2
∫ 1
2
∫ 6
1 3
f(x)dx = 1.8 and
f(x)dx = 3,,
∫ 4
f(x)dx
f(x)dx
f(x)dx+
∫ 8
5
(b)
(d)
∫ 3
0
∫ 2
f(x)dx−
f(x)dx+
∫ 3
2
∫ 3
−1 2
∫ 12
6
f(x)dx
f(x)dx = 3.2, find
f(x)dx = −7, and
∫ 4
∫ 5
2
f(x)dx.
f(x)dx = 2, find
f(x)dx
f(x)dx
0
0
3
1
∫ π
0
xsinxdx (b)
Answer to Exercise 4.3
∫ 5
1
e x ∫ 1
1+x dx (c)
2. (a) 1 (b) 42 (c) 2 (d) 4 3
(e) 317.5 (f) 3.75
5. (a)
∫ 3
f(x)dx (b)
∫ 2
f(x)dx (c)
∫ 1
(2x 2 −5x)dx (d)
0
1
f(x)dx (d)
∫ 3
∫ 4
f(x)dx.
√ xdx
0
0
0
−1 1
6. −1.4 7. −12
∫ 3
f(x)dx (e)
∫ 12
f(x)dx

MA111: Prepared by Dr.Archara Pacheenburawana 40
Exercise 4.4
1. Evaluate the definite integrals using the First Fundamental Theorem of Calculus.
(a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
(q)
(s)
(t)
∫ 3
−1
∫ 4
0
∫ 2
1
∫ 3
3
∫ π
π/2
∫ 9
1
∫ 9
8
∫ π/2
π/6
∫ 2
0
∫ 3
−2
∫ 2
0
x 5 dx
(b)
∫ 2
0
(2x−3)dx
∫
√ 4
xdx (d) ( √ x+3x)dx
∫
3
1
x dx (f) (x √ x+x −1/2 )dx
4 0
√
∫ π/2
x5 +2dx (h) 2sinxdx
secxtanxdx
(j)
0
0
∫ π
π/2
(2sinx−cosx)dx
∫
1
1
2x dx (l) (e x −e −x )dx
2 x dx (n)
(
x+ 2 )
sin 2 dx
x
(p)
|2x−3|dx
(r)
0
∫ 3
0
∫ √ 3
1
∫ 3π/4
0
(3e x −x 2 )dx
6
1+x 2 dx
|cosx|dx
{ −x, x ≥ 0
f(x)dx, where f(x) =
x 2 , x < 0
{ x
f(x)dx, where f(x) =
4 , 0 ≤ x < 1
x 5 , 1 ≤ x ≤ 2
2. Use the Second Fundamental Theorem of Calculus to find the derivative.
∫ x
∫ x √
(a) f(x) = (t 2 −3t+2)dt (b) g(x) = 1−2tdt
(c) g(y) =
(e) f(x) =
(g) h(x) =
(i) y =
∫ 1
0
∫ y
1−3x
2
∫ x 2
0
∫ 1/x
2
t 2 sintdt (d) F(x) =
(e −t2 +1)dt (f) f(x) =
arctantdt (h) y =
0
∫ 2
x
∫ −1
x
∫ √ x
u 3
∫ x 3
1+u du (j) y = 2
3
cos(t 2 )dt
cost
t
ln(t 2 +1)dt
dt
√ x
√
tsintdt

MA111: Prepared by Dr.Archara Pacheenburawana 41
Answer to Exercise 4.4
1. (a) 364
3
(b) −2 (c) 16 3
(d) 88
3
(e) 7 8
(f) 12
5
(g) 0 (h) 2 (i) Does not exist
(j) 3 (k) ln3 (l) e+e −1 −2 (m) 28
ln2
(q) 5 2
(r) 2− √ 2
2
(s) − 11 6
(t) 10.7
(n) 3e 3 −12 (o) π2
9 +2√ 3 (p) π 2
2. (a) f ′ (x) = x 2 −3x+2 (b) g ′ (x) = √ 1+2x (c) g ′ (y) = y 2 siny
(d) F ′ (x) = −cos(x 2 ) (e) f ′ (x) = (e −x4 +1)2x (f) f ′ (x) = −ln(x 2 +1)
(g) h ′ (x) = −arctan(1/x)/x 2 (h) y ′ = cos√ x
2x
(j) y ′ = 3x 7/2 sin(x 3 )−(sin √ x)/(2 4√ x)
Evaluate the definite integral, if it exists.
1.
3.
5.
7.
9.
11.
13.
15.
∫ 2
0
∫ 1
0
∫ 1
0
∫ 4
1
∫ 3
(x−1) 25 dx 2.
x 2 (1+2x 3 ) 5 dx 4.
Exercise 4.5
∫ 2
0
∫ 1
−1
∫ π
(i) y ′ = 3(1−3x)3
1+(1−3x) 2
x √ x 2 +1dx
x
(x 2 +1) 2 dx
4cosx
(sinx+1) 2 dx
cosπxdx 6.
π/2
√
1
1+ 1 ∫ π/2
x 2 x dx 8. cotxdx
dx
2x+3
0
∫ π/3
0
∫ 2
1
∫ a
0
10.
π/4
∫ 4
∫
sinx
13
cos 2 x dx 12.
x √ x−1dx 14.
x √ x 2 +a 2 dx (a > 0)
1
0
∫ e 4
e
x−1
√ x
dx
dx
√
3 (1+2x)
2
dx
x √ lnx
Answer to Exercise 4.5
1. 0 2. 5 3√
5−
1
3
3. 81
4
4. 0 5. 0 6. −2 7.
4 √ 2
3
− 5√ 5
12
8. 1 2 ln2 9. 1 2 ln3
10.
8
3
11. 1 12. 3 13. 16
15
14. 2 15. 1 3 (2√ 2−1)a 3

MA111: Prepared by Dr.Archara Pacheenburawana 42
1. Find the area of the shaded region.
Exercise 5.1
(a)
y
y = x 2 +1
−1 2
x
(b)
y
y = x 2 +2
y = −x
−1
2 x
(c)
y
y = 2−x 2
y = x
x
2. Find the area of the region bounded by the given curves.
(a) y = x 2 +3, y = x, −1 ≤ x ≤ 1
(b) y = x 3 , y = x 2 −1, 1 ≤ x ≤ 3
(c) y = x+1, y = 9−x 2 , −1 ≤ x ≤ 2

MA111: Prepared by Dr.Archara Pacheenburawana 43
(d) y = e x , y = x−1, −2 ≤ x ≤ 0
(e) y = x+1, y = (x−1) 2 , −1 ≤ x ≤ 2
(f) y = x 2 −1, y = 1−x, 0 ≤ x ≤ 2
(g) y = cosx, y = sin2x, 0 ≤ x ≤ π/2
(h) y = x 3 −1, y = 1−x, −2 ≤ x ≤ 2
3. Find the area of the region enclosed by the given curves.
(a) y = x 2 −1, y = 7−x 2
(b) y = x 2 +1, y = 3x−1
(c) y = x 3 , y = 3x+2
(d) y = x 3 , y = x 2
(e) y = x, y = 2−x, y = 0
(f) x = 3y, x = 2+y 2
(g) x = y, x = −y, x = 1
(h) y = x, y = 2, y = 6−x, y = 0
(i) x = 1−y 2 , x = y 2 −1
(j) y = cosx, y = 1−2x/π
(k) y = x 2 , y = 2/(x 2 +1)
(l) x = y 3 −y, x = 1−y 4
Answer to Exercise 5.1
1. (a) 6 (b) 40 3
(c) 9 2
2. (a) 20 (b) 40 (c) 19.5 (d) −e −2 (e) 31 (f) 3 (g) 1 (h) 29
3 3 6 2 2
3. (a) 64
3
(b) 1 6
(c) 27
4
(d) 1
12
(e) 1 (f) 1 6
(g) 1 (h) 8 (i) 1 8
(j) 2− π 2
(k) π − 2 3
(l) 8 5
Exercise 5.2
1. Findthevolumeofthesolidobtainedbyrotatingtheregionboundedbyy = 2−x, y =
0, and x = 0 (a) about the x-axis (b) about the line y = 3.
2. Find the volume of the solid obtained by rotating the region bounded by y = √ x, y =
2, and x = 0 (a) about the y-axis (b) about the line x = 4.
3. Find the volume of the solid obtained by rotating the region bounded by y = e x , x =
0, x = 2, and y = 0 (a) about the y-axis (b) about the line y = −2.

MA111: Prepared by Dr.Archara Pacheenburawana 44
4. Findthevolumeofthesolidobtainedbyrotatingtheregionboundedbyy = x 3 , y = 0,
and x = 1 (a) about the y-axis (b) about the x-axis.
5. The region R enclosed by the curves y = 3−x, the x-axis, and the y-axis. Find the
volume of the solid obtained by rotating the region R (a) about the y-axis, (b) about
the x-axis (c) about the line y = 3, (d) about the line y = −3, (e) about the line
x = 3, and (f) about the line x = −3.
6. The region R enclosed by the curves y = x 2 , y = 0, and x = 1. Find the volume of
the solid obtained by rotating the region R (a) about the y-axis, (b) about the x-axis,
(c) about the line x = 1, (d) about the line y = 1, (e) about the line x = −1, and (d)
about the line y = −1.
1. (a) 8π 3
4. (a) 2π 5
(b) 28π
3
(b) π 7
Answer to Exercise 5.2
2. (a) 32π
5
(b) 224π
15
( e
4
3. (a) 2πe 2 +2π (b) π
2 +4e2 − 9 )
2
5. (a) 9π (b) 9π (c) 18π (d) 36π (e) 18π (f) 36π
6. (a) π 2
(b) π 5
(c) π 6
(d) 7π
15
(e) 7π 6
(f) 13π
15
Exercise 5.3
1. Use the method of cylindrical shells to find the volume generated by rotating the
region bounded by the given curve about the y-axis.
(a) y = 1 , y = 0, x = 1, x = 2
x
(b) y = e −x2 , y = 0, x = 0, x = 1
(c) y 2 = x, x = 2y
2. Usethemethodofcylindricalshellstofindthevolumeofthesolidobtainedbyrotating
the region bounded by the given curve about the x-axis.
(a) x = 1+y 2 , x = 0, y = 1, y = 2
(b) y = x 2 , y = 9
(c) y = √ x, y = 0, x+y = 2
3. Use the appropriate method to find the volume generated by rotating the region
bounded by the given curve about the specified axis.
(a) y = √ x, x = 3, y = 0; about y-axis
(b) y = √ x, x = 5, y = 0; about x = 5
(c) y = 1 4 x3 +1, y = 1−x, x = 1; about y-axis
(d) x = y 2 , y = 1, x = 0; about x-axis
(e) x = y 2 , y = 2, x = 0; about y = 2

MA111: Prepared by Dr.Archara Pacheenburawana 45
Answer to Exercise 5.3
1. (a) 2π (b) π(1−1/e) (c) 64π/15 2. (a) 21π/2 (b) 1944π/5 (c) 5π/6
3. (a) 36√ 3
5
π (b) 40√ 5
3
π (c) 23π
30
(d) π 2
(e) 8π 3
Find the length of the curve.
Exercise 5.4
1. y = 1 3 (x2 +2) 3/2 , 0 ≤ x ≤ 1
2. y = x4
4 + 1
8x 2, 1 ≤ x ≤ 3
3. y = ln(secx), 0 ≤ x ≤ π/4
4. y = ln(1−x 2 ), 0 ≤ x ≤ 1 2
5. y = coshx, 0 ≤ x ≤ 1
Answer to Exercise 5.4
1.
4
3
2. 181
9
3. ln( √ 2+1) 4. ln3− 1 2
5. sinh1
Exercise 6.1
Evaluate the integral.
∫
∫
1. xe 2x dx 2. x 2 lnxdx
∫
∫
3. xsin4xdx 4. x 2 e −3x dx
∫
∫
5. x 2 cos3xdx 6. e x sin4xdx
∫
∫
7. (lnx) 2 dx 8. cosxcos2xdx
∫
∫
9. xsec 2 xdx 10. cosxln(sinx)dx
∫
∫
11. cos(lnx)dx 12. cos −1 xdx
∫
13. sin √ ∫ 1
xdx 14. xsin2xdx
15.
17.
∫ 1
0
∫ 4
1
xe −x dx 16.
ln √ xdx 18.
0
∫ 2
1
∫ 2
1
lnx
x 2 dx
x 4 (lnx) 2 dx

MA111: Prepared by Dr.Archara Pacheenburawana 46
Answer to Exercise 6.1
1.
1
2 xe2x − 1 4 e2x +C 2. 1 3 x2 lnx− 1 9 x3 +C 3. − 1 4 xcos4x+ 1 16 sin4x+C
4. − 1 3 x2 e −3x − 2 9 xe−3x − 2 27 e−3x +C 5. 1 3 x2 cos3x+ 2 9 xcos3x− 2 27 sin3x+C
6.
1
17 ex sin4x− 4
17 ex cos4x+C 7. x(lnx) 2 −2xlnx+2x+C
8.
2
sin2xcosx− 1 cos2xsinx+C 9. xtanx+ln|cosx|+C
3 3
10. sinxln(sinx)−sinx+C 11. 1 2 x[sin(lnx)+cos(lnx)]+C
12. xcos −1 x− √ 1−x 2 +C 13. −2 √ xcos √ x+2sin √ x+C 14. 1 4 sin2− 1 2 cos2
15. 1− 2 16. 1 − 1 ln2 17. 2ln4− 3 18. 3 e 2 2 2 2 (ln2)2 − 64 62
ln2+
25 125
Exercise 6.2
Evaluate the integral.
∫
∫
1. cosxsin 4 xdx 2. cos 2 xsinxdx
∫ 3π/4
∫
3. sin 5 xcos 3 xdx 4. cos 5 xsin 4 xdx
π/2
∫ π/2
∫
5. sin 2 3xdx 6. cos 2 xsin 2 xdx
0
∫
∫ π/4
7. (1−sin2x) 2 dx 8. sin 4 xcos 2 xdx
0
∫
9. sin 3 x √ ∫
cosxdx 10. cos 2 xtan 3 xdx
∫ ∫
1−sinx
11.
cosx dx
12. tan 2 xdx
∫
∫
13. sec 4 xdx 14. tan 5 xdx
∫
∫ π/4
15. tanxsec 3 xdx 16. tan 4 xsec 2 xdx
∫
17. tan 3 xsecxdx 18.
0
∫ π/3
0
tan 3 xsecxdx
∫ sec 2 ∫
x
π/2
19.
cotx dx 20. cot 2 xdx
π/6
∫
∫
21. cot 3 xcsc 3 xdx 22. cot 2 ωcsc 4 ωdω

MA111: Prepared by Dr.Archara Pacheenburawana 47
23.
25.
27.
∫
cscxdx 24.
∫ 1−tan 2 x
dx
sec 2 x
∫
26.
cos7θcos5θdθ 28.
∫ sinx
tanx dx
∫
sin5xsin2xdx
∫
sin4xcos5xdx
Answer to Exercise 6.2
1.
1
5 sin5 x+C 2. − 1 3 cos3 x+C 3. − 11
384
4. 1 5 sin5 x− 2 7 sin7 x+ 1 9 sin9 x+C
5.
π
4
6.
1
8 x− 1
32 sin4x+C 7. 3 2 x+cos2x− 1 8 sin4x+C 8. 1
192 (3π−4)
9. [ 2 7 cos3 x− 2 3 cosx]√ cosx+C 10. 1 2 cos2 x−ln|cosx|+C 11. ln(1+sinx)+C
12. tanx−x+C 13. 1 3 tan3 x+tanx+C 14. 1 4 sec4 x−tan 2 x+ln|secx|+C
15.
1
3 sec3 x+C 16. 1 5
17. 1 3 sec3 x−secx+C 18. 38
15
19. 1 2 tan2 x+C
20. √ 3− π 3
21. − 1 5 csc5 x+ 1 3 csc3 x+C 22. − 1 3 cot3 ω − 1 5 cot5 ω +C
23. ln|cscx−cotx|+C 24. sinx+C 25. 1 2 sin2x+C 26. 1 6 sin3x− 1
14 sin7x+C
27.
1
4 sin2θ+ 1 24 sin12θ+C 28. 1 2 cosx− 1
18 cos9x+C
Exercise 6.3
Evaluate the integral.
∫ 2
1.
t 3√ t 2 −1 dt
∫
1
3.
x 2√ 25−x dx
∫
2 x 2
5. √
x2 +9 dx
√
2
1
2. ∫
4. ∫
2
√
x2 −4 dx
dx
x √ x 2 +3 dx
6. ∫ √1−4x2
dx
∫ √x2
∫ √
9x2 −4
7. +16dx 8. dx
x
∫
x 2
∫
9.
(a 2 −x 2 ) dx 10. x
√ 3/2 x2 −7 dx
11.
13.
15.
∫ 3
0
∫ 2/3
∫
0
dx
√
9+x
2
12.
x 3√ 4−9x 2 dx 14.
1
√
9x2 +6x−8 dx
16. ∫
∫ 1
0
x √ x 2 +8dx
∫ √2x−x2
dx
dx
(x 2 +2x+2) 2

MA111: Prepared by Dr.Archara Pacheenburawana 48
17.
∫
e t√ 9−e 2t dt 18.
∫
dx
√
x2 +a 2 dx
Answer to Exercise 6.3
√
π
1. + √ 3
− 1 2. − 4−x2 25−x
24 8 4
2x +C 3. − 2
25x
+C 4. ( √ 1 3
)ln
∣
1
5.
3 (x2 +4) 3/2 −4 √ x 2 +4+C 6. 1 4 sin−1 (2x)+ 1x√ 1−4x
2 2 +C
√ 1
7. x√ x
2 2 x
+4+2ln
∣2 + x2 +4
2 ∣ +C 8. √ 9x 2 −4−2sec −1 ( 3x )+C 2
( √ x 2 +3− √ 3)
x
∣ +C
9.
13.
15.
x
√
a2 −x 2 −sin−1( x
a)
+C 10.
√
x2 −7+C 11. ln(1+ √ 2) 12. 1 3 (x2 +4) 3/2 +C
64
1215
14. 1 2 [sin−1 (x−1)+(x−1) √ 2x−x 2 ]+C
1
ln∣ √
∣
3 3x+1+ 9x2 +6x−8 ∣ +C 16.
1
2
[
tan −1 (x−1)+
]
(x+1)
+C
(x 2 +2x+2)
[
1
17. √ 2 e
t
9−e 2t +9sin −1 ( et)]
+C 18. ln(x+ √ x
3 2 +a 2 )+C
Exercise 6.4
Evaluate the integral.
∫ ∫
x−5
1.
x 2 −1 dx
2. 6x
x 2 −x+2 dx
∫
∫
x+1
3.
x 2 −x−6 dx 4. −x+5
x 3 −x 2 −2x dx
∫ x 3 ∫
+x+2
−3x−1
5.
x 2 +2x−8 dx 6. x 3 −x dx
∫ ∫
2x+3
7.
(x+2) dx
8. x−1
2 x 3 +4x 2 +4x dx
∫
∫
x+4
x+2
9.
x 3 +3x 2 +2x dx 10. x 3 +x dx
∫ ∫
4x−2
11.
x 4 −1 dx
12. 3x 2 −6
x 2 −x−2 dx
∫
∫
2x+3
x 2
13.
x 2 +2x+1 dx 14. +2x+1
dx
x 3 +x
∫
4x 2 ∫
+3
15.
x 3 +x 2 +x dx 16. 3x 3 +1
x 3 −x 2 +x−1 dx

MA111: Prepared by Dr.Archara Pacheenburawana 49
Answer to Exercise 6.4
1. 3ln|x+1|−2ln|x−1|+C 2. 2ln|x+1|+4ln|x−2|+C
3.
1
5 ln|x+2|+ 4 5 ln|x−3|+C 4. 2ln|x+1|+ 1 2 ln|x−2|− 5 2 ln|x|+C
5. 11ln|x+4|+2ln|x−2|+ 1 2 x2 −2x+C
6. ln|x+1|−2ln|x−1|+ln|x|+C 7. 2ln|x+2|−(x+2) −1 +C
1
8. ln|x+2|− 3 4 2 (x+2)−1 − 1 ln|x|+C 9. ln|x+2|−3ln|x+1|+2ln|x|+C
4
10. −ln(x 2 +1)+tan −1 x+2ln|x|+C
11.
3
2 ln|x+1|+ 1 2 ln|x−1|−ln(x2 +1)+tan −1 x+C
12. 3x+ln|x+1|+2ln|x−2|+C 13. 2ln|x+1|−(x+1) −1 +C
14. 2tan −1 x+ln|x|+C 15. 3ln|x|+ 1 2 ln|x2 +x+1|− 7 √
3
tan −1 (2x+1 √
3
)+C
16. 3x+2ln|x−1|+ 1 2 ln(x2 +1)−2tan −1 x+C
Exercise 6.5
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
1.
3.
5.
7.
9.
11.
13.
15.
17.
∫ ∞
1
∫ ∞
0
∫ ∞
−∞
∫ ∞
0
∫ ∞
1
∫ ∞
1
∫ 0
−1
∫ 3
−2
∫ 2
−2
1
(3x+1) 2 dx
e −x dx 4.
xe −x2 dx 6.
cosxdx 8.
lnx
x dx
2. ∫ −1
−∞
∫ ∞
−∞
∫ ∞
0
∫ 1
−∞
10. ∫ ∞
−∞
lnx
x 2 dx 12. ∫ 3
0
1
√ 2−w
dw
x 3 dx
1
(x+2)(x+3) dx
xe 2x dx
x
1+x 2 dx
1
√ x
dx
∫
1
π/4
x dx 14. csc 2 tdt
2
1
x 4 dx
0
∫ π
16. secxdx
∫
1
2
x 2 −1 dx 18. z 2 lnzdz
0
0

MA111: Prepared by Dr.Archara Pacheenburawana 50
Answer to Exercise 6.5
1.
1
2
2. D 3. 1 4. D 5. 0 6. −ln 2 3
7. D 8. e2 4
9. D 10. D 11. 1
12. 2 √ 3 13. D 14. D 15. D 16. D 17. D 18. 8 3 ln2− 8 9
Exercise 7.1
1. List the first five terms of the sequence.
(a) a n = 1−(0.2) n (b) a n = n+1
3n−1
(c) a n = 3(−1)n
(e)
{
sin nπ 2
n!
}
(d) {2·4·6·····(2n)}
(f) a 1 = 1, a n = 1
1+a n
2. Findaformulaforthegeneraltermterma n ofthesequence, assuming thatthepattern
of the first few terms continuous.
(a) { 1
2 , 1 4 , 1 8 , 1
16 ,...} (b) { 1
2 , 1 4 , 1 6 , 1 8 ,...}
(c) {2,7,12,17,...} (d) { − 1 4 , 2 9 ,− 3
16 , 4 25 ,...}
(e) { 1,− 2 3 , 4 9 ,− 8
27 ,...} (f) {0,2,0,2,0,2,...}
3. Determine whether the sequence converges of diverges. If it converges, find the limit.
(a) a n = n(n−1) (b) a n = n+1
3n−1 √
(c) a n = 3+5n2
n
(d) a
n+n 2 n =
1+ √ n
(e) a n = 2n
3 n+1 (f) a n =
(g) a n = (−1)n−1 n
n 2 +1
(i) a n = 2+cosnπ
{ } 3+(−1)
n
(k)
n
{ 2 } ln(n 2 )
(m)
n
(o) { √ n+2− √ n}
n
1+ √ n
(h) a n = sin(nπ/2)
(j) {arctan2n}
{ } n!
(l)
(n+2)!
(n) {(−1) n sin(1/n)}
{ } ln(2+e n )
(p)
3n
(q) a n = n2 −n
(r) a n = ln(n+1)−lnn

MA111: Prepared by Dr.Archara Pacheenburawana 51
(s) a n = cos2 n
2 n (t) a n = (1+3n) 1/n
(u) a n = 1 n + 2 2 n +···+ n (v) a 2 n 2
n = ncosn
n 2 +1
(w) a n = n!
(x) a
2 n n = (−3)n
n!
Answer to Selected Exercise 7.1
1. (a) 0.8,0.96,0.992,0.9984,0.99968 (c) −3, 3 2 ,−1, 1,− 1 (e) {1,0,−1,0,1,...}
2 8 40
2. (a) a n = 1/2 n (c) 5n−3 (e) ( − 2 3
) n−1
3. (a) D (c) 5 (e) 0 (g) 0 (i) D (k) 0 (m) 0 (o) 0 (q) 0 (s) 0 (u) 1 2
(w) D
Exercise 7.2
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1. 4+ 8 + 16 + 32 +··· 2. −2+ 5 − 25 + 125 −···
5 25 125 2 8 32
∞∑
3. 5 ( 2 n−1
∞∑ (−3)
3) n−1
4.
5.
7.
9.
11.
13.
15.
17.
n=1
∞∑
3 −n 8 n+1 6.
n=1
∞∑
n=1
∞∑
n=1
1
n(n+2)
n
√
1+n
2
8.
10.
∞∑
arctann 12.
n=1
∞∑ [ (
2
1
) n ( )
4 +3 −
1 n ]
5
n=1
14.
∞∑
( 1
n − 1 )
n−1
16.
∞∑ ( e
) n+1
π
18.
n=2
n=1
n=1
n=1
4 n
∞∑ n
n+5
∞∑
[2(0.1) n +(0.2) n ]
n=1
∞∑ 3 n +2 n
n=1
6 n
∞∑ n
ln
n+1
n=1
∞∑ n−5
n+2
n=1
∞∑ n!
100 n
n=1
∞∑
(
3
(n−1) − 3 )
2 n 2
n=2

MA111: Prepared by Dr.Archara Pacheenburawana 52
Answer to Exercise 7.2
1.
20
3
2. D 3. 15 4. 1 7
5. D 6. D 7. 3 4
8. 17
36
9. D 10. 3 2
11. D 12. D
13.
31
6
14. D 15. −1 16. D 17.
e 2
π(π−e)
18. 3
n=1
Exercise 7.3
Determine whether the series converges or diverges.
∞∑ 1
∞∑ 1
∞∑
1.
2.
3. ne −n
n 4 3n+1
n=1
n=1
n=1
∞∑ 1
4.
5. 1+ 1 n 1.0001 8 + 1 27 + 1
64 +··· 6. ∑ ∞
5−2 √ n
n 3
n=5
n=1
∞∑
∞∑ n
∞∑ 1
7. ne −n2 8.
9.
n 2 +1
nlnn
n=1
n=1
n=2
∞∑ arctann
∞∑ 1
∞∑ −2
10.
11.
12. √
1+n 2 n 2 +2n+2 n+2
13.
16.
19.
22.
25.
28.
31.
34.
37.
40.
∞∑
n=2
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
7
4n+2
1
n 2 +n+1
14.
17.
3
n2 n 20.
3+cosn
3 n 23.
1
1+ √ n
26.
n+1
n2 n 29.
n
n 2 +2n+3
32.
n!
n 100 35.
n+3
n 2√ n
1
(2n)!
38.
41.
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
3
(4+3n) 7/6 15.
5
2+3 n 18.
1
√
n(n+1)(n+2)
21.
n
√
n5 +4
n 2 +1
n 4 +1
1
n!
1
n √ n+1
n 3
(2n)!
n 2
n!
24.
27.
30.
33.
36.
39.
sin2n
n 2 42.
n=1
∞∑
n=2
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
∞∑
n=1
ne −3n2
n+1
n 2
n 2 +1
n 3 −1
2 n
1+3 n
1+n+n 2
√
1+n2 +n 6
( 1
sin
n)
8 n
n!
n
n+200
4n 3 +3n
n 5 −4n 2 +1
n(−3) n
4 n−1

MA111: Prepared by Dr.Archara Pacheenburawana 53
43.
46.
49.
52.
∞∑
n=1
∞∑
n=1
∞∑
n=1
10 n
(n+1)4 2n+1 44.
n n
3 1+3n 47.
n n
(2n)!
50.
∞∑ n!
(−10) n 45.
∞∑
( ) n 2 n
+1
2n 2 +1
48.
∞∑ 4 n +n
n!
51.
n=1
n=1
n=1
∞∑ cos(nπ/3)
n!
∞∑ 1
2+sin 2 n
∞∑
( n
3n+2
1
1·2 + 1
2·3 + 1
3·4 + 1
4·5 +··· 53. 1 3 + 2 3 2 + 3 3 3 + 4 3 4 +···
54. 1+ 1
2 √ 2 + 1
3 √ 3 + 1
4 √ 4 +··· 55. ∞
∑
56.
2
1·3·4 + 3
2·4·5 + 4
3·5·6 + 5
4·6·7 +···
n=1
Answer to Exercise 7.3
n=1
n=1
n=1
2·4·6·····(2n)
n!
1. C 2. D 3. C 4. C 5. C 6. C 7. C 8. D 9. D 10. C 11. C 12. D
13. D 14. C 15. C 16. C 17. C 18. D 19. C 20. C 21. D 22. C 23. C
24. C 25. D 26. C 27. D 28. C 29. C 30. D 31. D 32. C 33. C 34. D
35. C 36. D 37. C 38. C 39. C 40. C 41. C 42. C 43. C 44. D 45. C
46. D 47. C 48. C 49. C 50. C 51. C 52. C 53. C 54. C 55. D 56. C
Exercise 7.4
1. Find the Maclaurin series for the function.
(a) e −x (b) e ax (c) cosπx
(d) sinπx (e) ln(1+x)
1
(f)
1+x
(g) coshx (h) sinhx (i) xsinx
(j) xe x
2. Find the Taylor series of the function f(x) centered at the given value of a.
(a) e x ; a = 1 (b) e −x ; a = ln2 (c) 1 x ; a = −1
1
(d)
x+2 ; a = 3 (e) sinπx; a = 1 (f) cosx; a = π 2
2
(g) lnx; a = 1
(h) lnx; a = e
) n

MA111: Prepared by Dr.Archara Pacheenburawana 54
1. (a)
(i)
2. (a)
(g)
∞∑ (−1) n
x n (c)
n!
n=0
∞∑
n=0
∞∑
n=0
(−1) n
(2n+1)! x2n+2
e
n! (x−1)n (c)
Answer to Selected Exercise 7.4
∞∑ (−1) n π 2k
x 2n (e)
(2n)!
n=0
∞∑ (−1) n−1
(x−1) n
n
n=0
∞∑
(−1)(x+1) n (e)
n=0
∞∑ (−1) n+1
x n (g)
n
n=0
∞∑ (−1) n π 2n
n=0
(2n)!
∞∑
n=0
(
x− 1 ) 2n
2
1
(2n)! x2n